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Chapter 4 90
4-1 Congruent Figures
1. Underline the correct word to complete the sentence.
A polygon is a two-dimensional figure with two / three or more segments that meet exactly at their endpoints.
2. Cross out the figure(s) that are NOT polygons.
Vocabulary Builder
congruent (adjective) kahng GROO unt
Main Idea: Congruent fi gures have the same size and shape.
Related Word: congruence (noun)
Use Your Vocabulary
3. Circle the triangles that appear to be congruent.
Write T for true or F for false.
4. Congruent angles have different measures.
5. A prism and its net are congruent figures.
6. The corresponding sides of congruent figures have the same measure.
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Problem 1
Key Concept Congruent Figures
Congruent polygons have congruent corresponding parts—their matching sides and angles. When you name congruent polygons, you must list corresponding vertices in the same order.
7. Use the figures at the right to complete each congruence statement.
A B
C
ABCD EFGH
D
EF
G H
AB > BC > CD > DA >
/A > /B > /C > /D >
Problem 2
W Y S M K V
W
Y
91 Lesson 4-1
Using Congruent Parts
Got It? If kWYS O kMKV , what are the congruent corresponding parts?
8. Use the diagram at the right. Draw an arrow from each vertex of the first triangle to the corresponding vertex of the second triangle.
9. Use the diagram from Exercise 8 to complete each congruence statement.
Sides WY > YS > WS >
Angles /W > /Y > /S >
Finding Congruent Parts
Got It? Suppose that kWYS O kMKV. If m/W 5 62 and m/Y 5 35, what is mlV ? Explain.
Use the congruent triangles at the right.
10. Use the given information to label thetriangles. Remember to write corresponding vertices in order.
11. Complete each congruence statement.
/W >
/Y >
/S >
12. Use the Triangle Angle-Sum theorem.
m/S 1 m 1 m 5 180, so m/S 5 180 2 ( 1 ), or .
13. Complete.
Since /S > and m/S 5 , m/V 5 .
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Problem 4
Problem 3
B
D
CA
Theorem 4–1 Third Angles Theorem
B
D
E FC
A
AB
C
ED
Chapter 4 92
Finding Congruent Triangles
Got It? Is kABD O kCBD? Justify your answer.
14. Underline the correct word to complete the sentence.
To prove two triangles congruent, show that all adjacent / corresponding parts are congruent.
15. Circle the name(s) for nACD.
acute isosceles right scalene
16. Cross out the congruence statements that are NOT supported by the information in the figure.
AD > CD BD > BD AB > CB
/A > /C /ABD > /CBD /ADB > /CDB
17. You need congruence statements to prove two triangles congruent, so you
can / cannot prove that nABD > nCBD.
Use kABC and kDEF above.
18. If m/A 5 74, then m/D 5 .
19. If m/B 5 44, then m/E 5 .
20. If m/C 5 62, then m/F 5 .
Proving Triangles Congruent
Got It? Given: lA O lD, AE O DC ,
EB O CB, BA O BD
Prove: kAEB O kDCB
21. You are given four pairs of congruent parts. Circle the additional information you need to prove the triangles congruent.
A third pair A second pair A third pair of congruent of congruent of congruent sides angles angles
Theorem
If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent.
If . . .
/A > /D and /B > /E
Then . . .
/C > /F
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Need toreview
0 2 4 6 8 10
Lesson Check
93 Lesson 4-1
Check off the vocabulary words that you understand.
congruent polygons
Rate how well you can identify congruent polygons.
22. Complete the steps of the proof.
1) AE > , EB > , BA > 1) Given
2) /A > 2) Given
3) /ABE > 3) Vertical angles are congruent.
4) /E > 4) Third Angles Theorem
5) nAEB > 5) Definition of > triangles
• Do you UNDERSTAND?
If each angle in one triangle is congruent to its corresponding angle in another triangle, are the two triangles congruent? Explain.
23. Underline the correct word to complete the sentence.
To disprove a conjecture, you need one / two / many counterexample(s).
24. An equilateral triangle has three congruent sides and three 608angles. Circle the equilateral triangles below.
25. Use your answers to Exercise 24 to answer the question.
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____________________________________________________________________
____________________________________________________________________
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C
B
A
Chapter 4 94
Triangle Congruence by SSS and SAS4-2
1. Use the diagram at the right. Find each.
included angle between AB and CA
included side between /A and /C
included angle between BC and CA
included side between /B and /C
included angle between BC and AB
included side between /B and /A
Vocabulary Builder
postulate (noun) PAHS chuh lit
Definition: A postulate is a statement that is accepted as true without being proven true.
Main Idea: In geometry, you use what you know to be true to prove new things true. The statements that you accept as true without proof are called postulates or axioms.
Use Your Vocabulary
2. Underline the correct word to complete the sentence.
You can use properties, postulates, and previously proven theorems as
reasons / statements in a proof.
3. Multiple Choice What is a postulate?
a convincing argument using deductive reasoning
a conjecture or statement that you can prove true
a statement accepted as true without being proven true
a conclusion reached by using inductive reasoning
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d.Postulate 4–1 Side-Side-Side (SSS) Postulate
Postulate 4-1 Side-Side-Side (SSS) Postulate
If the three sides of one triangle are congruent to the three sides of another triangle, then the two triangles are congruent.
4. Use the figures at the right to complete the sentence.
If AB > DE, BC > EF, and AC > , then nABC > n .
A
B
C
D
E
F
Problem 1
B
C
D
F
A C
B
R
T
S
D F
E
Postulate 4–2 Side-Angle-Side (SAS) Postulate
95 Lesson 4-2
Using SSS
Got It? Given: BC O BF, CD O FD Prove: kBCD O kBFD
5. You know two pairs of sides that are congruent. What else do you need to prove the triangles congruent by SSS?
_______________________________________________________________________
6. The triangles share side .
7. Complete the steps of the proof.
Statement Reason
1) BC > 1) Given
2) CD > 2) Given
3) BD > 3) Reflexive Property of >
4) nBCD > 4) SSS
Postulate 4–2 Side-Angle-Side (SAS) Postulate
If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.
Use the figures below to complete each statement.
8. nDEF > by SAS.
9. nABC > by SSS.
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Problem 3 Identifying Congruent Triangles
Got It? Would you use SSS or SAS to prove the triangles below congruent? Explain.
Complete each statement with SSS or SAS.
17. Use 9 if you have three pairs of sides congruent.
18. Use 9 if you have two pairs of sides and the included angle congruent.
Write T for true or F for false.
19. The diagram shows congruence of three sides.
20. In the triangle on the left, the marked angle is the included angle of the side
with two marks and the side with three marks.
21. In the triangle on the right, the marked angle is the included angle of the side
with two marks and the side with three marks.
BN
L E
Problem 2
Chapter 4 96
Using SAS
Got It? What other information do you need to prove kLEB O kBNL by SAS?
10. Circle the angles that are marked congruent in the diagram.
/EBL /ELB /NBL /NLB
11. Circle the sides that form the angles that are marked congruent in the diagram.
BE BL BN LB LE LN
12. Complete each congruence statements.
LB > /BLE >
Underline the correct word(s) to complete each sentence.
13. Proving nLEB > nBNL by SAS requires one / two pair(s) of congruent sides and
one / two pair(s) of congruent angles.
14. The diagram shows congruency of zero / one / two pair(s) of congruent sides and
zero / one / two pair(s) of congruent angles.
15. To prove the triangles congruent by SAS, you still need zero / one / two pair(s) of
congruent sides and zero / one / two pair(s) of congruent angles .
16. To prove the triangles congruent, you need to prove and congruent.
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d. 22. Would you use SSS or SAS to prove the triangles congruent? Explain.
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
Lesson Check
Error Analysis Your friend thinks that the triangles below are congruent by SAS. Is your friend correct? Explain.
23. Are two pairs of corresponding sides congruent? Yes / No
24. Is there a pair of congruent angles? Yes / No
25. Are the congruent angles the included angles between the corresponding congruent sides? Yes / No
26. Are the triangles congruent by SAS? Explain.
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
• Do you UNDERSTAND?
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97 Lesson 4-2
Check off the vocabulary words that you understand.
congruent corresponding
Rate how well you can use SSS and SAS to prove triangles congruent.
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Vocabulary
Review
A C
B
Z X
Y
Chapter 4 98
Triangle Congruenceby ASA and AAS4-3
1. Cross out the figure(s) that are NOT triangle(s).
2. A triangle is a polygon with sides.
3. A triangle with a right angle is called a(n)
obtuse / right / scalene triangle.
Vocabulary Builder
corresponding (adjective) kawr uh SPAHN ding
Other Word Forms: correspond (verb); correspondence (noun)
Definition: Corresponding means similar in position, purpose, or form.
Math Usage: Congruent figures have congruent corresponding parts.
Use Your Vocabulary
Draw a line from each part of kABC in Column A to the corresponding part of kXYZ in Column B.
Column A Column B
4. BC /Z
5. /A /Y
6. AB YZ
7. /C /X
8. AC XY
9. /B XZ
A
B C
corresponding sides
Y
X
Z
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Problem 1
Postulate 4–3 Angle-Side-Angle (ASA) Postulate
F
IN
C
T
AO
H
G
AF
E
D
B
C
Using ASA
Got It? Which two triangles are congruent by ASA? Explain.
11. Name the triangles. List the vertices in corresponding order: list the vertex with the one arc first, the vertex with the two arcs second, and the third vertex last.
12. /G > / > /
13. /H > / > /
14. HG > >
15. The congruent sides that are included between congruent angles are
and .
16. Write a congruence statement. Justify your reasoning.
n > n
_______________________________________________________________________
_______________________________________________________________________
10. Explain how the ASA Postulate is different from the SAS Postulate.
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_______________________________________________________________________
_______________________________________________________________________
Postulate
If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.
If . . .
/A > /D, AC > DF, /C > /F
Then . . .
nABC > nDEF
99 Lesson 4-3
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Problem 2
Problem 3
B
A
E
DC
F
B
A
C
ED
Theorem 4–2 Angle-Angle-Side (AAS) Theorem
P
S Q
R
Given
are right angles./B and /
Given
BA ˘
All right angles are congruent.
/B ̆
ASA Postulate
nABC ˘
Given
/CAB ˘ /
Chapter 4 100
Writing a Proof Using ASA
Got It? Given: lCAB O lDAE, BA O EA, lB and lE are right angles Prove: kABC O kAED
17. Complete the flow chart to prove nABC > nAED.
Writing a Proof Using AAS
Got It? Given: lS O lQ, RP bisects lSRQ Prove: kSRP O kQRP
19. How do you know which angles in the diagram are corresponding angles?
_______________________________________________________________________
_______________________________________________________________________
20. Complete the statements to provenSRP > nQRP.
Statements Reasons
1) /S > 1) Given
2) RP bisects 2) Given
3) /SRP > 3) Definition of an angle bisector
4) RP > 4) Reflexive Property of Congruence
5) nSRP > 5) AAS
18. The nonincluded congruent sides of nABC and nDEF are and .
If . . .
/A > /D, /B > /E, AC > DF
Then . . .
nABC > nDEF
Theorem
If two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of another triangle, then the two triangles are congruent.
HSM11GEMC_0403.indd 100 4/9/09 3:30:56 PM
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Lesson Check
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Math Success
Problem 4
AR
P I
S
F
• Do you UNDERSTAND?
Reasoning Suppose lE O lI and FE O GI . What else must you know in order to prove kFDE and kGHI are congruent by ASA? By AAS?
24. Label the diagram at the right.
25. To prove the triangles congruent by ASA, what do you need?
_________________________________________
_________________________________________
26. To prove the triangles congruent by AAS, what do you need?
_______________________________________________________________________
_______________________________________________________________________
27. If you want to use ASA, / and / must also be congruent.
28. If you want to use AAS, / and / must also be congruent.
Check off the vocabulary words that you understand.
included nonincluded corresponding
Rate how well you can use ASA and AAS.
Determining Whether Triangles Are Congruent
Got It? Are kPAR and kSIR congruent? Explain.
21. The congruence marks show that /A > and PR > .
22. What other corresponding congruent parts exist? Explain.
_______________________________________________________________________
23. Are nPAR and nSIR congruent? If so, what theorem proves them congruent?
_______________________________________________________________________
101 Lesson 4-3
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Using Corresponding Parts of Congruent Triangles4-4
Chapter 4 102
Underline the correct word(s) to complete each sentence.
1. The Reflexive Property of Congruence states that any geometric figure is
congruent / similar to itself.
2. The Reflexive Property of Equality states that any quantity is
equal to / greater than / less than itself.
3. Circle the expressions that illustrate the Reflexive Property of Equality.
a 5 a If AB 5 2, then 2 5 AB.
3(x 1 y) 5 3x 1 3y 5 1 c 5 5 1 c
4. Circle the expressions that illustrate the Reflexive Property of Congruence.
If /A > /B, then /B > /A. If CD > LM and LM > XY, then CD > XY.
/ABC > /ABC CD > CD
Vocabulary Builder
proof (noun) proof
Related Word: prove (verb)
Definition: A proof is convincing evidence that a statement or theory is true.
Math Usage: A proof is a convincing argument that uses deductive reasoning.
Use Your Vocabulary
Complete each statement with proof or prove.
5. In geometry, a 9 uses definitions, postulates, and theorems to prove theorems.
6. No one can 9 how our universe started.
7. He can 9 when he bought the computer because he has a receipt.
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Problem 1
B D
CA
E
103 Lesson 4-4
Proving Parts of Triangles Congruent
Got It? Given: BA O DA, CA O EA
Prove: lC O lE
9. Name four ways you can use congruent parts of two triangles to prove that the triangles are congruent.
10. To prove triangles are congruent when you know two pairs of congruent
corresponding sides, you can use or .
Underline the correct word to complete the sentence.
11. The Given states and the diagram shows that there are one / two / three pairs of congruent sides.
12. Give a reason for each statement of the proof.
Statements Reasons
1) BA > DA 1)
2) CA > EA 2)
3) /CAB > /EAD 3)
4) nCAB > nEAD 4)
5) /C > /E 5)
8. Complete the steps in the proof.
Given: AB > AD, BC > DC, /D > /B, /DAC > /BAC
Prove: nABC > nADC
Statements Reasons
1) AB > BC > 1) Given
2) AC > 2) Reflexive Property of >
3) /D > /DAC > 3) Given
4) /DCA > 4) Third Angles Theorem
5) nABC > 5) Definition of > triangles
A C
D
B
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Problem 2
Lesson Check
B
M A
C
E MA
T
Corresponding parts of ˘ triangles are ˘.
/AMB ˘
Given
M is the midpoint of
Given
AB ˘
Definition of midpoint
AM ˘
BM ˘
Reflexive Property of ˘
SSS Theorem
nAMB ˘
.
• Do you know HOW?
Name the postulate or theorem that you can use to show the triangles are congruent. Then explain why EA O MA.
14. Circle the angles that are marked congruent.
/E /ETA /M /EAT /MTA
15. Circle the sides that are marked congruent.
ET MT EA MA AT
16. Circle the sides that are congruent by the Reflexive Property of Congruence.
ET and MT EA and MA AT and AT
17. Underline the correct postulate or theorem to complete the sentence.
nEAT > nMAT by SAS / AAS / ASA / SSS .
18. Now explain why EA > MA.
_______________________________________________________________________
Proving Triangle Parts Congruent to Measure Distance
Got It? Given: AB O AC, M is the midpoint of BC
Prove: lAMB O lAMC
13. Use the flow chart to complete the proof.
Chapter 4 104
HSM11GEMC_0404.indd 104 3/12/09 9:41:27 AM
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Math Success
Lesson Check
L
K
N
HM
Check off the vocabulary words that you understand.
congruent corresponding proof
Rate how well you can use congruent triangles.
• Do you UNDERSTAND?
Error Analysis Find and correct the error(s) in the proof.
Given: KH > NH, /L > /M Prove: H is the midpoint of LM .
Proof: KH > NH because it is given. /L > /M because it is given./KHL > /NHM because vertical angles are congruent. So, nKHL > nMHNby ASA Postulate. Since corresponding parts of congruent triangles are congruent,LH > MH . By the defi nition of midpoint, H is the midpoint of LM .
Place a ✓ in the box if the statement is correct. Place an ✗ if it is incorrect.
19. /KHL > /NHM because vertical angels are congruent.
20. nKHL > nMHN by ASA Postulate.
Underline the correct word to complete each sentence.
21. When you name congruent triangles, you must name corresponding vertices in
a different / the same order.
22. To use the ASA Postulate, you need two pairs of congruent angles and a pair of
included / nonincluded congruent sides.
23. To use the AAS Theorem, you need two pairs of congruent angles and a pair of
included / nonincluded congruent sides.
24. Identify the error(s) in the proof.
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
25. Correct the error(s) in the proof.
_______________________________________________________________________
105 Lesson 4-4
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Vocabulary
Equilateral Isosceles Right
Chapter 4 106
Isosceles and Equilateral Triangles4-5
ReviewUnderline the correct word to complete each sentence.
1. An equilateral triangle has two/ three congruent sides.
2. An equilateral triangle has acute / obtuse angles.
3. Circle the equilateral triangle.
8
55
4
3 5
5
5 5
Vocabulary Builder
isosceles (adjective) eye SAHS uh leez
Related Words: equilateral, scalene
Definition A triangle is isosceles if it has two congruent sides.
Main Idea: The angles and sides of isosceles triangles have special relationships.
Use Your Vocabulary
4. Use the triangles below. Write the letter of each triangle in the correct circle(s) at the right.
isosceles
E
BD
C
A
HSM11GEMC_0405.indd 106 3/12/09 9:47:14 AM
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Vertex Angle
Legs
Base
Base Angles
C
BDA
C
BDA
R V S
WU
TProblem 1
Theorems 4–3, 4–4, 4–5
Vertex Angle
Legs
Base
Base Angles
107 Lesson 4-5
Theorem 4-3 Isosceles Triangle Theorem
If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
Theorem 4-4 Converse of Isosceles Triangle Theorem
If two angles of a triangle are congruent, then the sides opposite those angles are congruent.
Theorem 4-5
If a line bisects the vertex angle of an isosceles triangle, then the line is also the perpendicular bisector of the base.
5. If PQ > RQ in nPQR, then / > / .
6. Underline the correct theorem number to complete the sentence.
Th e theorem illustrated below is Th eorem 4–3 / 4–4 / 4–5 .
If . . . Th en . . .
Using the Isosceles Triangle Theorems
Got It? Is lWVS congruent to lS? Is TR congruent to TS? Explain.
7. The markings show that WV > .
8. Is /WVS > /S? Explain.
_______________________________________________________________________
_______________________________________________________________________
9. Is/R > /S? Explain.
_______________________________________________________________________
_______________________________________________________________________
10. Is TR > TS? Explain.
_______________________________________________________________________
_______________________________________________________________________
HSM11GEMC_0405.indd 107 3/12/09 9:47:18 AM
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Problem 2
Problem 3
Y
X Z
A
D xB
C
Y
X Z
D C
A E B
Chapter 4 108
Using Algebra
Got It? Suppose mlA 5 27. What is the value of x?
11. Since CB > , nABC is isosceles.
12. Since nABC is isosceles, m/A 5 m/ 5 .
13. Since BD bisects the vertex of an isosceles triangle, BD '
and m/BDC 5 .
14. Use the justifications below to find the value of x.
m/ 1m/BDC1x 5 180 Triangle Angle-Sum Theorem
1 1x 5 180 Substitute.
1x 5 180 Simplify.
x 5 Subtract 117 from each side.
Corollaries to Theorems 4–3 and 4–4
Corollary to Theorem 4-3
If a triangle is equilateral, then the triangle is equiangular.
Corollary to Theorem 4-4
If a triangle is equiangular, then the triangle is equilateral.
15. Underline the correct number to complete the sentence.
Th e corollary illustrated below is Corollary to Th eorem 4-3 / 4-4 .
If . . . Th en . . .
Finding Angle Measures
Got It? Suppose the triangles at the right are isosceles triangles, where lADE, lDEC, and lECB are vertex angles. If the vertex angles each have a measure of 58, what are mlA and mlBCD?
16. Which triangles are congruent by the Side-Angle-Side Theorem?
17. Which angles are congruent by the Isosceles Triangle Theorem?
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What is the relationship between sides and angles for each type of triangle?
isosceles equilateral
Complete.
22. An isosceles triangle has congruent sides.
23. An equilateral triangle has congruent sides.
Complete each statement with congruent, isosceles, or equilateral.
24. The Isosceles Triangle Theorem states that the angles opposite the congruent sides are 9.
25. Equilateral triangles are also 9 triangles.
26. The sides and angles of an 9 triangle are 9.
• Do you UNDERSTAND?
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109 Lesson 4-5
18. By the Triangle Angle-Sum Theorem, m/A 1 58 1 m/DEA 5 .
19. Solve for m/A.
20. Since > /ECD, m/ECD 5 .
21. Using the Angle Addition Postulate, m/BCD 5 58 1 m/ECD 5 .
Check off the vocabulary words that you understand.
corollary legs of an isosceles triangle base of an isosceles triangle
vertex angle of an isosceles triangle base angles of an isosceles triangle
Rate how well you understand isosceles and equilateral triangles.
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Chapter 4 110
Congruence in Right Triangles 4-6
Write T for true or F for false.
1. Segments that are congruent have the same length.
2. Polygons that are congruent have the same shape but are not always the same size.
3. In congruent figures, corresponding angles have the same measure.
Vocabulary Builder
hypotenuse (noun) hy PAH tuh noos
Related Word: leg
Definition: Th e hypotenuse is the side opposite the right angle in a right triangle.
Main Idea: Th e hypotenuse is the longest side in a right triangle.
Use Your Vocabulary
Underline the correct word(s) to complete each sentence.
4. One side of a right triangle is / is not a hypotenuse.
5. A right triangle has one / two / three legs.
6. The length of the hypotenuse is always equal to / greater than / less than the lengths of the legs.
Use the triangles at the right for Exercises 7 and 8.
7. Cross out the side that is NOT a hypotenuse.
BC AB GH FD
8. Circle the leg(s).
AC AB HI ED
legle
g
hypotenuse
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Theorem 4-6 Hypotenuse-Leg (HL) Theorem and Conditions
Theorem If . . . Then . . .
If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and leg of another right triangle, then the triangles are congruent.
nPQR and nXYZ are right triangles,
PR > XZ , and PQ > XY
nPQR > nXYZ
Q
P
R Y
X
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111 Lesson 4-6
You can prove that two triangles are congruent without having to show that all corresponding parts are congruent. In this lesson, you will prove right triangles congruent by using one pair of right angles, a pair of hypotenuses, and a pair of legs.
Use the information in the Take Note for Exercises 10–12.
10. How do the triangles in the Take Note meet the first condition in Exercise 9? Explain.
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11. How do the triangles in the Take Note meet the second condition in Exercise 9? Explain.
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12. How do the triangles in the Take Note meet the third condition in Exercise 9? Explain.
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9. To use the HL Theorem, the triangles must meet three conditions. Complete each sentence with right or congruent.
There are two 9 triangles.
The triangles have 9 hypotenuses.
There is one pair of 9 legs.
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Problem 1
Problem 2
Given
are right angles./PRS and /
Given
SP >
Definition of right triangle
are right triangles.nPRS and n
Reflexive Prop. of >
PR >
HL Theorem
nPRS > n
Chapter 4 112
Using the HL Theorem
Got It? Given: /PRS and /RPQ are right angles, SP O QR Prove: kPRS O kRPQ
13. Complete each step of the proof.
Writing a Proof Using the HL Theorem
Got It? Given: CD O EA, AD is the perpendicular bisector of CE Prove: kCBD O kEBA
14. Circle what you know because AD is the perpendicular bisector of CE .
/CBD and /EBA are right angles. /CBD and /EBA are acute angles.
B is the midpoint of AD. B is the midpoint of CE .
15. Circle the congruent legs.
AB CB DB EB
16. Write the hypotenuse of each triangle.
nCBD nEBA
17. Complete the proof.
Statements Reasons
1) CD > 1) Given
2) /CBD and / are right /s. 2) Definition of ' bisector
3) nCBD and n are right ns. 3) Definition of right n
4) CB > 4) Definition of ' bisector
5) nCBD > 5) HL Theorem
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Lesson Check
M
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113 Lesson 4-6
Check off the vocabulary words that you understand.
hypotenuse legs of a right triangle
Rate how well you can use the Hypotenuse-Leg (HL) Th eorem.
• Do you UNDERSTAND?
Error Analysis Your classmate says that there is not enough information to determine whether the two triangles at the right are congruent. Is your classmate correct? Explain.
Write T for true or F for false.
18. There are three right angles.
19. There are two right triangles.
20. There are two congruent hypotenuses.
21. There are no congruent legs.
22. You need to use the Reflexive Property of Congruence.
23. LJ > LJ is given.
24. Do you always need three congruent corresponding parts to prove triangles congruent? Explain.
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25. Is your classmate correct? Explain.
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Chapter 4 114
4-7 Congruence in Overlapping Triangles
1. Circle the common side of nABC and nADC .
AB AC AD BC
2. Circle the common side of nXWZ and nYWZ .
WZ WX WY ZY
3. Circle the common side of nRST and nRPT .
RP RS RT ST
Vocabulary Builder
overlapping (adjective) oh vur LAP ing
Other Word Form: overlap (noun)
Definition: Overlapping events or figures have parts in common.
Math Usage: Two or more figures with common regions are overlapping figures.
Use Your Vocabulary
Circle the common regions of the overlapping figures in the diagram at the right.
4. nFGD and nCBE
nABG nACF nEHD nGHB
5. nBEC and nHED
nBEC nGBH nGDF nHED
6. nACF and nABG
nABG nACF nGBH nEHD
7. nACF and nGBH
nABG nACF nGBH nHED
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115 Lesson 4-7
Identifying Common Parts
Got It? What is the common side in kABD and kDCA?
8. Separate and redraw nABD and nDCA.
9. You drew twice, so the common side is .
Using Common Parts
Got It? Given: kACD > kBDC
Prove: CE > DE
10. Use the information in the problem to complete theproblem-solving model below.
11. Use the justifications below to complete each statement.
Statements Reasons
1) nACD > 1) Given
2) AC > 2) Corresponding parts of > triangles are >.
3) /A > 3) Corresponding parts of > triangles are >.
4) > /BED 4) Vertical angles are congruent.
5) > nBED 5) Angle-Angle-Side (AAS) Theorem
6) CE > 6) Corresponding parts of > triangles are >.
12. How could you use the Converse of the Isosceles Triangle Theorem to prove CE > DE?
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Know Need Plan
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Problem 4
Problem 3
P
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PSQ RSQ
PST RST and QPS QRS
QPT QRT
PS RS ST ST and QS QS
QT QT
PT RT PQ RQ
A D
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A D
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Chapter 4 116
Separating Overlapping Triangles
Got It? Given: lCAD O lEAD, lC O lE
Prove: BD O FD
14. Circle the angles that are vertical angles.
/ADB /ADC /ADE /ADF /BDC /FDE
15. Mark the angles that you know are congruent in each pair of separated triangles below.
16. Which triangles are congruent by AAS? Explain.
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Using Two Pairs of Triangles
Got It? Given: PS O RS, /PSQ O /RSQ
Prove: kQPT O kQRT
13. Give the reason for each statement in the proof.
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117 Lesson 4-7
• Do you UNDERSTAND?
In the figure at the right, which pair of triangles could you prove congruent first in order to prove that kACD O kCAB? Explain.
19. Is the hypotenuse of nACD congruent to the hypotenuse of nCAB? Explain.
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20. What else do you need to prove right angles congruent using HL?
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21. Which triangles can you prove congruent to find this? Explain.
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17. Which triangles are congruent by ASA? Explain.
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18. How can you prove BD > FD?
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Check off the vocabulary words that you understand.
congruent corresponding overlapping
Rate how well you can identify congruent overlapping triangles.
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